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Theorem oppgplusfval 15064
Description: Value of the addition operation of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Fan Zheng, 26-Jun-2016.)
Hypotheses
Ref Expression
oppgval.2  |-  .+  =  ( +g  `  R )
oppgval.3  |-  O  =  (oppg
`  R )
oppgplusfval.4  |-  .+b  =  ( +g  `  O )
Assertion
Ref Expression
oppgplusfval  |-  .+b  = tpos  .+

Proof of Theorem oppgplusfval
StepHypRef Expression
1 oppgplusfval.4 . 2  |-  .+b  =  ( +g  `  O )
2 oppgval.2 . . . . . . 7  |-  .+  =  ( +g  `  R )
3 fvex 5675 . . . . . . 7  |-  ( +g  `  R )  e.  _V
42, 3eqeltri 2450 . . . . . 6  |-  .+  e.  _V
54tposex 6442 . . . . 5  |- tpos  .+  e.  _V
6 plusgid 13484 . . . . . 6  |-  +g  = Slot  ( +g  `  ndx )
76setsid 13428 . . . . 5  |-  ( ( R  e.  _V  /\ tpos  .+  e.  _V )  -> tpos  .+  =  ( +g  `  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
) )
85, 7mpan2 653 . . . 4  |-  ( R  e.  _V  -> tpos  .+  =  ( +g  `  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
) )
9 oppgval.3 . . . . . 6  |-  O  =  (oppg
`  R )
102, 9oppgval 15063 . . . . 5  |-  O  =  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
1110fveq2i 5664 . . . 4  |-  ( +g  `  O )  =  ( +g  `  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
)
128, 11syl6reqr 2431 . . 3  |-  ( R  e.  _V  ->  ( +g  `  O )  = tpos  .+  )
13 tpos0 6438 . . . . 5  |- tpos  (/)  =  (/)
146str0 13425 . . . . 5  |-  (/)  =  ( +g  `  (/) )
1513, 14eqtr2i 2401 . . . 4  |-  ( +g  `  (/) )  = tpos  (/)
16 reldmsets 13411 . . . . . . 7  |-  Rel  dom sSet
1716ovprc1 6041 . . . . . 6  |-  ( -.  R  e.  _V  ->  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )  =  (/) )
1810, 17syl5eq 2424 . . . . 5  |-  ( -.  R  e.  _V  ->  O  =  (/) )
1918fveq2d 5665 . . . 4  |-  ( -.  R  e.  _V  ->  ( +g  `  O )  =  ( +g  `  (/) ) )
20 fvprc 5655 . . . . . 6  |-  ( -.  R  e.  _V  ->  ( +g  `  R )  =  (/) )
212, 20syl5eq 2424 . . . . 5  |-  ( -.  R  e.  _V  ->  .+  =  (/) )
2221tposeqd 6411 . . . 4  |-  ( -.  R  e.  _V  -> tpos  .+  = tpos 
(/) )
2315, 19, 223eqtr4a 2438 . . 3  |-  ( -.  R  e.  _V  ->  ( +g  `  O )  = tpos  .+  )
2412, 23pm2.61i 158 . 2  |-  ( +g  `  O )  = tpos  .+
251, 24eqtri 2400 1  |-  .+b  = tpos  .+
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1649    e. wcel 1717   _Vcvv 2892   (/)c0 3564   <.cop 3753   ` cfv 5387  (class class class)co 6013  tpos ctpos 6407   ndxcnx 13386   sSet csts 13387   +g cplusg 13449  oppgcoppg 15061
This theorem is referenced by:  oppgplus  15065
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-i2m1 8984  ax-1ne0 8985  ax-rrecex 8988  ax-cnre 8989
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-tpos 6408  df-recs 6562  df-rdg 6597  df-nn 9926  df-2 9983  df-ndx 13392  df-slot 13393  df-sets 13395  df-plusg 13462  df-oppg 15062
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